This proposal is about several topics in arithmetic geometry. Riemann Roch formulas for coherent Euler characteristics will be studied, with applications to Iwasawa theory and capacity theory. The proposal also has to do with quadratic invariants of coherent sheaves and of unit groups. Various methods from number theory and algebraic geometry will be used to show that arithmetic groups can be generated by elements of small height or by subgroups of small rank. Finally, the inverse problem for deformation rings will be considered, as well as the liftability of group actions on curves in positive characteristic.
This proposal is about symmetry and its applications in number theory and arithmetic geometry. The exploitation of symmetry to understand the solutions of equations and the geometry of objects has been a basic theme in mathematics. In this proposal, symmetry will be used to show that complicated algebraic systems arising from number theory and geometry can in many cases be described in simple and compact ways. Symmetries will also be used to distinguish when such systems are fundamentally different from one another and when they can or cannot be extended beyond their original scope.
This project concerned several topics in number theory and geometry. An underlying theme of the project was to find better ways to measure the size or shape of mathematical objects. A classical measure of the shape of a three-dimensional object, for example, is to count how many holes it has. In mathematical terms, this pertains to the Euler characteristic of the object. Euler characteristics turn up in many different forms. A central goal of this project was to study them in the context of solutions of algebraic equations. This led to proving some new Riemann Roch formulas for Euler characteristics. This work in turn led to applying the same techniques to a different area of mathematics, Iwasawa theory. Iwasawa theory has to do measuring how the complexity of a sequence of equations grows. Another measure of the size and shape of an object comes from the theory of electrostatics, and is called the capacity of the object. The capacity of a piece of metal, for example, measures the equilibrium energy of one unit of electrical charge on the piece. The theory of capacity is useful in number theory because it can predict when one can construct polynomial functions with certain boundedness properties. These functions enter into important applications in cryptography, since one can use them to break codes given enough partial information about secret keys. A new form of capacity was developed during the course of this project, and its applications to cryptography are currently being explored.
